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Intermode interactions in one-dimensional nonlinear periodic structures havebeen studied by many authors, starting with the classical work by Fermi, Pasta,and Ulam (FPU) in the middle of the last century. However, symmetry selectionrules for the energy transfer between nonlinear vibrational modes of differentsymmetry, which lead to the possibility of excitation of some bushes of suchmodes, were not revealed. Each bush determines an exact solution of nonlineardynamical equations of the considered system. The collection of modes of agiven bush does not change in time, while there is a continuous energy exchangebetween these modes. Bushes of nonlinear normal modes (NNMs) are constructedwith the aid of group-theoretical methods and therefore they can exist for thecase of large amplitude atomic vibrations and for any type of interatomicinteractions. In most publications, bushes of NNMs or similar dynamical objectsin one-dimensional systems are investigated under periodic boundary conditions.In this paper, we present a detailed study of the bushes of NNMs in monoatomicchains for the case of fixed boundary conditions, which sheds light on a seriesof new properties of the intermode interactions in such systems. We prove sometheorems that justify a method for constructing bushes of NNMs by continuationof conventional normal modes to the case of large atomic oscillations. Ourstudy was carried out for FPU chains, for the chains with the Lennard-Jonesinteratomic potential, as well as for the carbon chains (carbynes) in theframework of the density functional theory. For one-dimensional bushes(Rosenberg nonlinear normal modes), the amplitude-frequency diagrams arepresented and the possibility of their modulational instability is brieflydiscussed. We also argue in favor of the fact that our methods and main resultsare valid for any monoatomic chain.

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends